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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-inftyexpiinv | Structured version Visualization version GIF version |
Description: Utility theorem for the inverse of +∞ei. (Contributed by BJ, 22-Jun-2019.) This utility theorem is irrelevant and should generally not be used. (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-inftyexpiinv | ⊢ (𝐴 ∈ (-π(,]π) → (1st ‘(+∞ei‘𝐴)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1 4796 | . . . 4 ⊢ (𝑥 = 𝐴 → 〈𝑥, ℂ〉 = 〈𝐴, ℂ〉) | |
2 | df-bj-inftyexpi 34483 | . . . 4 ⊢ +∞ei = (𝑥 ∈ (-π(,]π) ↦ 〈𝑥, ℂ〉) | |
3 | opex 5348 | . . . 4 ⊢ 〈𝐴, ℂ〉 ∈ V | |
4 | 1, 2, 3 | fvmpt 6762 | . . 3 ⊢ (𝐴 ∈ (-π(,]π) → (+∞ei‘𝐴) = 〈𝐴, ℂ〉) |
5 | 4 | fveq2d 6668 | . 2 ⊢ (𝐴 ∈ (-π(,]π) → (1st ‘(+∞ei‘𝐴)) = (1st ‘〈𝐴, ℂ〉)) |
6 | cnex 10612 | . . 3 ⊢ ℂ ∈ V | |
7 | op1stg 7695 | . . 3 ⊢ ((𝐴 ∈ (-π(,]π) ∧ ℂ ∈ V) → (1st ‘〈𝐴, ℂ〉) = 𝐴) | |
8 | 6, 7 | mpan2 689 | . 2 ⊢ (𝐴 ∈ (-π(,]π) → (1st ‘〈𝐴, ℂ〉) = 𝐴) |
9 | 5, 8 | eqtrd 2856 | 1 ⊢ (𝐴 ∈ (-π(,]π) → (1st ‘(+∞ei‘𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 〈cop 4566 ‘cfv 6349 (class class class)co 7150 1st c1st 7681 ℂcc 10529 -cneg 10865 (,]cioc 12733 πcpi 15414 +∞eicinftyexpi 34482 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fv 6357 df-1st 7683 df-bj-inftyexpi 34483 |
This theorem is referenced by: bj-inftyexpiinj 34485 |
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